Reconstructing Loads in Nanoplates from Dynamic Data
It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&q...
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| Format: | Article |
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MDPI AG
2023-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/4/398 |
| _version_ | 1827745902170734592 |
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| author | Alexandre Kawano Antonino Morassi |
| author_facet | Alexandre Kawano Antonino Morassi |
| author_sort | Alexandre Kawano |
| collection | DOAJ |
| description | It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>f</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup></semantics></math></inline-formula> of the transverse load <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup><msub><mi>g</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mi>f</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>g</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup></semantics></math></inline-formula> is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>f</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup></semantics></math></inline-formula>. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction. |
| first_indexed | 2024-03-11T05:15:01Z |
| format | Article |
| id | doaj.art-6e24bff016064020857af6f11b5f618e |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T05:15:01Z |
| publishDate | 2023-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-6e24bff016064020857af6f11b5f618e2023-11-17T18:19:48ZengMDPI AGAxioms2075-16802023-04-0112439810.3390/axioms12040398Reconstructing Loads in Nanoplates from Dynamic DataAlexandre Kawano0Antonino Morassi1Escola Politécnica, University of São Paulo, Av. Prof. Luciano Gualberto, 380-Butantã, São Paulo 05508-010, BrazilPolytechnic Department of Engineering and Architecture, University of Udine, Via Cotonificio 114, 33100 Udine, ItalyIt was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>f</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup></semantics></math></inline-formula> of the transverse load <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup><msub><mi>g</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msub><mi>f</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>g</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup></semantics></math></inline-formula> is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mrow><mo>{</mo><msub><mi>f</mi><mi>m</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>M</mi></msubsup></semantics></math></inline-formula>. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction.https://www.mdpi.com/2075-1680/12/4/398inverse problemsload reconstructionnanoplatesstrain-gradient elasticitylinear dynamics |
| spellingShingle | Alexandre Kawano Antonino Morassi Reconstructing Loads in Nanoplates from Dynamic Data Axioms inverse problems load reconstruction nanoplates strain-gradient elasticity linear dynamics |
| title | Reconstructing Loads in Nanoplates from Dynamic Data |
| title_full | Reconstructing Loads in Nanoplates from Dynamic Data |
| title_fullStr | Reconstructing Loads in Nanoplates from Dynamic Data |
| title_full_unstemmed | Reconstructing Loads in Nanoplates from Dynamic Data |
| title_short | Reconstructing Loads in Nanoplates from Dynamic Data |
| title_sort | reconstructing loads in nanoplates from dynamic data |
| topic | inverse problems load reconstruction nanoplates strain-gradient elasticity linear dynamics |
| url | https://www.mdpi.com/2075-1680/12/4/398 |
| work_keys_str_mv | AT alexandrekawano reconstructingloadsinnanoplatesfromdynamicdata AT antoninomorassi reconstructingloadsinnanoplatesfromdynamicdata |